**1. Standard Form
**

A second order linear differential equation is a differential equation of the form

(Here A,B,C and D are certain prescribed functions of x.)

As in the case of first order linear equations, in any interval where A(x) ≠ 0,
we can replace such an equation

by an equivalent one in **standard form**:

where

**2. Homogeneous vs. Non-homogeneous Linear Differential
Equations
**

In the development that follows it will be important to distinguish between the case when the right hand

side of

is zero or non-zero. We shall say that a second order
linear ODE is **homogeneous** if it can be written in

the form

otherwise (if g(x) ≠ 0) we shall say that it is
**non-homogeneous**. Note that this terminology is completely

unrelated to homogeneous equations of degree zero (the topic of the preceding
lecture).

**3. Differential Operator Notation
**

Consider the general second order linear differential equation

We shall often write differential equations like this as

where L is the** linear differential operator**

That is to say, L is the operator that acts on a function Ø by

**4. General Theorems
**

The following theorem tells us the conditions for the existence and uniqueness of solutions of a second order

linear differential equation.

Theorem 14.1. If the functions p, q and g are continuous on an open interval I R containing the point

, then in some interval about there exists a unique solution to the differential equation

satisfying the prescribed initial conditions

Note how this theorem is analogous to the corresponding
theorem for first order linear ODE's. Note also

that the conditions for existence and uniqueness are fairly lax - all we require
is the continuity of the

functions p, q, and g around a given initial point. Finally, we note that the
form of the initial conditions

involves the specification of **both** y(x) and its derivative y'(x) at an initial point

.

I should also point out that the preceding theorem does not address the issue of
how to construct a solution

of a second order linear ODE. Indeed, the actual construction of solutions to
second order linear ODE is

sufficiently complicated to that we shall spend 90% of the remaining lectures on
techniques of solution. The

next two theorems at least tell us the basic ingredients for a general solution
of a second order linear ODE.

Theorem 14.2. (The Superposition Principle) If y = (x) and y = (x) are two
solutions of the differential

equation

then any linear combination

of (x) and (x), where
and are constants, is also
a solution of (14.12).

Proof.

The fact that a linear combination of solutions of a**
linear, homogeneous differential equation **is

also a solution is extremely important. The theory of linear homogeneous
equations, including differential

equations involving higher derivatives depends strongly on the superposition
principle

Example 14.3.

and

are both solutions of

It is easy to check that any linear combination of and
is also a solution.

Example 14.4.

and

are both solutions of

However, it is easy to check that
is not a solution of (14.20). The reason for
this lies in

the fact that (14.20) is not linear.

Given two solutions and of a second order linear homogeneous differential
equation

we can construct an infinite number of other solutions

by letting and run through R. The following question
then arises: are** all **the solutions of (14.21)

capable of being expressed in form (14.22) for some choice of and ?

This will not always be the case; and so we shall say that two solutions and
form a **fundamental set**

of solutions to (14.21) if every solution of (14.21) can be expressed as a
linear combination of
and

.

Theorem 14.5. If p and q are continuous on an open interval I = ( α, β) and if
and are solutions of

the differential equation

satisfying

at every point x∈ I, then any other solution of (14.23) on
the interval I can be expressed uniquely as a

linear combination of and .

Proof.

Let and be two given solutions on an interval I and let Y be an any other
solution on I. Choose a

point ∈I. From our basic uniqueness and existence theorem (Theorem 3.2), we know that
there is only

solution y(x) of (14.23) such that

namely, Y (x). Therefore if we can show that a solution of the form

satisfies the initial conditions (14.25), then we must
have and so Y (x) is a linear

combination of (x) and (x).

Thus, we now seek to define constants and so that
these initial conditions can be matched. We thus

set

This is just a series of two equations with two unknowns. Solving the first equation for yields

Plugging this into the second equation yields

or

or

Plugging this expression for into (14.27) yields

Thus, we can solve for and whenever the denominator

does not vanish. Thus, so long as and satisfy
(14.23) we can always express any solution as a linear

combination of and .

Remark: The quantity

is called the **Wronskian** of and .

Example 14.6. Show that

and

are form a set of fundamental solutions to the differential equation

We simply have to check that the Wronskian does not vanish:

Since the Wronskian does not vanish, and must be linearly independent.